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Error analysis of exponential integrators for oscillatory second-order differential equations

Volker Grimm, Marlis Hochbruck
2006 Journal of Physics A: Mathematical and General  
In this paper we analyse a family of exponential integrators for secondorder differential equations in which high-frequency oscillations in the solution are generated by a linear part.  ...  Our convergence analysis generalises known results on the mollified impulse method by García-Archilla, Sanz-Serna and Skeel [6] and on Gautschi-type exponential integrators [12, 13] .  ...  This prevents the method from being of order two as a one-step method in the form (3) . A different one-step formulation of the Gautschi-type method is given in [13] .  ... 
doi:10.1088/0305-4470/39/19/s10 fatcat:nkagweoghrdxdmacnuhoeqjowa

Numerical solution of nonlinear wave equations in stratified dispersive media

Ch. Karle, J. Schweitzer, M. Hochbruck, E.W. Laedke, K.H. Spatschek
2006 Journal of Computational Physics  
A new variant of the Gautschi-type integrator for reducing the number of time steps is proposed as a fast solver for such nonlinear wave-equations.  ...  In order to reduce the number of spatial grid points, a physically motivated quasi-envelope approach (QEA) is introduced.  ...  Note that this switch requires to recompute the differential operator and hence the matrix operators required for the Gautschi-type integrator.  ... 
doi:10.1016/j.jcp.2005.11.024 fatcat:kgjmnqsuh5avnpe2dfln46seqi

Special methods for problems whose oscillatory solution is damped

Beny Neta
1989 Applied Mathematics and Computation  
This paper complements the results of Pratt and others on exponentialfitted methods and those of Gautschi, Neta, van der Wouwen and others on trigonometrically-fitted methods.  ...  This paper introduces methods tailored especially for prcblems whose solution behaves like eXX, where h is complex. The shallow water equations with topography admit such solution.  ...  Acknowledgment The author would like to thank the NPS Foundation Research Program for its support of this research.  ... 
doi:10.1016/0096-3003(89)90116-1 fatcat:awznvazjp5alznjce7wde3wvji

Page 6519 of Mathematical Reviews Vol. , Issue 2000i [page]

2000 Mathematical Reviews  
65L70 65M12 65M20 Hochbruck, Marlis (D-TBNG-MI; Tiibingen) ; Lubich, Christian (D-TBNG-MI; Tiibingen) A Gautschi-type method for oscillatory second-order differential equations.  ...  From the summary: “Many orbit problems in celestial mechan- ics are described by (nonstiff) initial-value problems for second- order ordinary differential equations of the form y” = f(y).  ... 

Important aspects of geometric numerical integration

Ernst Hairer
2005 Journal of Scientific Computing  
Using a modulated Fourier expansion, much insight can be gained for methods applied to problems where the high oscillations stem from a linear part of the vector field and where only one (or a few) high  ...  At the example of Hamiltonian differential equations, geometric properties of the flow are discussed that are only preserved by special numerical integrators (such as symplectic and/or symmetric methods  ...  The author is grateful to the organizers for providing the stimulating atmosphere at this workshop.  ... 
doi:10.1007/bf02728983 fatcat:lomtimn3wberlotb3z2enu4e3m

Block Trigonometrically Fitted Backward Differentiation Formula for the Initial Value Problem with Oscillating Solutions

R.I. Abdulganiy, O.A. Akinfenwa, S.A. Okunuga
2022 Nigerian Journal of Basic and Applied Sciences  
The BTBDF is a product of three discrete formulas which are obtained from a continuous second derivative trigonometrically fitted method (CSDTFM).  ...  first-order Initial Value Problems (IVPs) with oscillatory results.  ...  ACKNOWLEDGEMENTS The authors would like to thank the anonymous referees whose comments and suggestions greatly improve the manuscript.  ... 
doi:10.4314/njbas.v29i1.1 fatcat:4qksfm4t3nf5niow724jgw6nwe

Families of methods for ordinary differential equations based on trigonometric polynomials

B. Neta, C.H. Ford
1984 Journal of Computational and Applied Mathematics  
We consider the construction of methods based on trigonometric polynomials for the initial value problems whose solutions are known to be periodic.  ...  Gautschi [4] developed Adams and Strrmer type methods. In our paper we construct Nystrrm's and Milne-Simpson's type methods.  ...  Acknowledgment The authors would like to thank Prof. George J. Fix for the invitation to Carnegie Mellon University and the use of the computer.  ... 
doi:10.1016/0377-0427(84)90066-9 fatcat:63d6xnrdnvevrho3ubm7txtxyu

Symmetric high order Gautschi-type exponential wave integrators pseudospectral method for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime [article]

Yan Wang, Xiaofei Zhao
2016 arXiv   pre-print
A group of high order Gautschi-type exponential wave integrators (EWIs) Fourier pseudospectral method are proposed and analyzed for solving the nonlinear Klein-Gordon equation (KGE) in the nonrelativistic  ...  With the Fourier pseudospectral method to discretize the KGE in space, we propose a group of EWIs with designed Gautschi's type quadratures for the temporal integrations, which can offer any intended even  ...  Acknowledgments This work is supported by the French ANR project MOONRISE ANR-14-CE23-0007-01.  ... 
arXiv:1611.01550v1 fatcat:bkco75jug5afxld4lkr7qshlda

Asymptotic consistent exponential-type integrators for Klein-Gordon-Schrödinger systems from relativistic to non-relativistic regimes

Simon Baumstark, Georgia Kokkala, Katharina Schratz
2018 Electronic Transactions on Numerical Analysis  
In particular, our first-and second-order exponential-type integrators are asymptotically consistent in the sense of asymptotically converging to the corresponding decoupled free Schrödinger limit system  ...  In this paper we propose asymptotic consistent exponential-type integrators for the Klein-Gordon-Schrödinger system.  ...  Even more suitable so-called Gautschi-type methods, which are especially designed for the numerical solution of oscillatory second-order differential equations (see, e.g., [2, 17, 19] ), do not allow  ... 
doi:10.1553/etna_vol48s63 fatcat:xpwplgh4ijgltjcoduyoxhdia4

On the numerical quadrature of highly-oscillating integrals I: Fourier transforms

A. Iserles
2004 IMA Journal of Numerical Analysis  
Our analysis is applied to modified Magnus methods for highly-oscillatory ordinary differential equations.  ...  We focus our analysis on Filon-type quadrature and analyse its behaviour in a range of frequency regimes for integrals of the form h 0 f (x)e iωx w(x)dx, where h > 0 is small and |ω| large.  ...  Acknowledgements The author is indebted to many colleagues who shared their expertise on numerical quadrature and acted as a sounding board for the ideas in this paper, in particular to Brad Baxter, Ilan  ... 
doi:10.1093/imanum/24.3.365 fatcat:e4f74sn5vzbhtiwyajohfyux7u

Numerical Comparison between Different Lie-Group Methods for Solving Linear Oscillatory ODEs [chapter]

Fasma Diele, Stefania Ragni
2002 Lecture Notes in Computer Science  
Moreover we introduce a new method based on the Cayley map and provide some numerical tests in order to compare the different approaches.  ...  In this paper we deal with high oscillatory systems and numerical methods for the approximation of their solutions.  ...  Successively in [3] Hochbruch and Lubich introduced the Gautschi-type methods that reduce to solve exactly the equation with constant inhomogeneity.  ... 
doi:10.1007/3-540-47789-6_50 fatcat:mjft6kt42rervhabsgpx3oj3w4

A parallel implementation of a two-dimensional fluid laser–plasma integrator for stratified plasma–vacuum systems

Ch. Karle, J. Schweitzer, M. Hochbruck, K.H. Spatschek
2008 Journal of Computational Physics  
The use of a one-step version of the Gautschi-type integrator, being accurate of second order independent of the highest frequencies arising in the system, is proposed.  ...  This splitting allows to evaluate the matrix functions arising in the Gautschi-type method by onedimensional Fourier transforms.  ...  Fig. 6 . 6 For the Gautschi-type method (solid) and the leap-frog method (dashed) the runtime between outputs (i.e. 228 time-steps, except for vacuum steps with the Gautschi-type method) is shown.  ... 
doi:10.1016/j.jcp.2008.04.024 fatcat:lyhht5nxunh4pmednyxaj34bgu

Spectral Methods for Time-Dependent Variable-Coefficient PDE Based on Block Gaussian Quadrature [chapter]

James V. Lambers
2010 Lecture Notes in Computational Science and Engineering  
In this paper, we review the most effective type of KSS method, that relies on block Gaussian quadrature, and compare its performance to that of Krylov subspace methods from the literature.  ...  Krylov subspace spectral (KSS) methods have previously been applied to the variable-coefficient heat equation and wave equation, as well as systems of coupled equations such as Maxwell's equations, and  ...  second-order wave equation, in a way that preserves the order of accuracy achieved for the wave equation.  ... 
doi:10.1007/978-3-642-15337-2_41 fatcat:xsswd4ttuvfzzjfzp2idqtbxie

A Gautschi-type method for oscillatory second-order differential equations

Marlis Hochbruck, Christian Lubich
1999 Numerische Mathematik  
We study a numerical method for second-order differential equations in which high-frequency oscillations are generated by a linear part. For example, semilinear wave equations are of this type.  ...  The numerical scheme is based on the requirement that it solves linear problems with constant inhomogeneity exactly.  ...  Acknowledgements We are grateful to Gerhard Wanner for helpful comments on the presentation.  ... 
doi:10.1007/s002110050456 fatcat:3schvcowuvf6rfpw7ps5uetlgm

author index 10 volumes

2005 Journal of Computational and Applied Mathematics  
Vandewalle, A note on wave number dependence of wavelet matrix compression for integral equations with oscillatory kernel 172 (2004) 233-246 Ismail, M., Problem 4.  ...  Tocino, A., Mean-square stability of second-order Runge-Kutta methods for stochastic differential equations 175 (2004) 355-367 Tokarzewski, S., see Gilewicz, J. 178 (2005) 247-253 Torney, D.C., Bayesian  ... 
doi:10.1016/s0377-0427(05)00273-6 fatcat:4hnauo7pn5b2hksb3pel3ew244
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